Frobenius algebra

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In mathematics, a Frobenius algebra is an associative algebra A defined over a field K equipped with a special kind of bilinear form

[ f: A \times A \rightarrow K, ]

then called a Frobenius form of the algebra. It is required to have the following properties:

(Associativity)

For all [ a,b,c \in A: f(ab,c)=f(a,bc) ]

(Non-degeneracy)

[ f(a,b)=0 \;\; \forall \; a \in A \Longrightarrow b=0 ]

Example

Any matrix algebra defined over a field K is a Frobenius algebra with Frobenius form given as

b(A,B) = tr(AB),

where tr denotes the matrix trace.

Applications

Frobenius algebras occur in the representation theory of algebras. More recently it has been seen that they play an important role in the algebraic treatment and axiomatic foundation of topological quantum field theory.

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